1. The question provides information about a past UPSEE mathematics exam from 2006 containing 40 multiple choice questions covering topics in complex numbers, trigonometry, geometry, and coordinate geometry. 2. For each question, four possible answers (a, b, c, or d) are provided and test-takers must indicate the correct answer in their answer book. 3. The document contains 40 multiple choice questions testing a range of mathematical concepts and skills.

1. UPSEE–PAST PAPERS
MATHEMATICS - UNSOLVED PAPER – 2006

2. SECTION
 Single Correct Answer Type
 There are five parts in this question. Four choices are given for each part and one of them is
correct. Indicate you choice of the correct answer for each part in your answer-book by
writing the letter (a), (b), (c) or (d) whichever is appropriate

3. 01 Problem
If z1, z2 are any two complex numbers, then :
a. | z1 z2 | | z1 | | z2 |
b. | z1 z2 | | z1 | | z2 |
c. | z1 z2 | | z1 | | z2 |
d. | z1 z2 | | z1 | | z2 |

4. 02 Problem
If z = x + iy is a variable complex number such that arg , z 1 then :
z 1 4
a. x2 – y2 – 2x = 1
b. x2 + y2 – 2x = 1
c. x2 + y2 – 2y = 1
d. x2 + y2 + 2x = 1

5. 03 Problem
If arithmetic mean of tow positive numbers is A, their geometric mean is G and
harmonic mean is H, then H is equal to :
a. G2 / A
b. A2 / G2
c. A / G2
d. G / A2

6. 04 Problem
The sun of n terms of two arithmetic series are in the ratio 2n + 3 : 6n + 5, then
the ratio of their 13th terms is :
a. 53 : 155
b. 27 : 87
c. 29 : 83
d. 31 : 89

7. 05 Problem
If , , are the roots of the equation x3 + x + 1 = 0 , then the value of
3 3 3 is :
a. 0
b. 3
c. - 3
d. - 1

8. 06 Problem
0 0 1
If , then A-1 is :
A 0 1 0
1 0 0
a. - A
b. A
c. 1
d. none of these

9. 07 Problem
6 8 5
4 2 3
If A = 9 7 1 is the sum of symmetric matrix B and skew-symmetric
matrix C, then B is :
6 6 7
6 2 5
a. 9 7 1
0 2 2
2 5 2
b.
2 2 0
6 6 7
c. 6 2 5
7 5 1
0 6 2
d. 2 0 2
2 2 0

10. 08 Problem
1 1
If A = 1 1
, then A100 is equal to :
a. 2100 A
b. 299A
c. 100 A
d. 299A

11. 09 Problem
1 1 1 x 0 x
If 1 2 2 y 3 , then y is equal to :
1 3 1 z 4 z
0
a. 1
1
1
b. 2
3
5
2
c.
1
1
2
d. 3

12. 10 Problem
If y = cos2 x + sec2 x, then :
a. y 2
b. y 1
c. y 2
d. 1 < y < 2

13. 11 Problem
1 1
If x 2 cos , then x2 is equal to :
2 x3
a. sin 3
b. 2 sin 3
c. cos 3
d. 2 cos 3

14. 12 Problem
If sin + cosec = 2, the value of sin10 + cosec10 is :
a. 2
b. 210
c. 29
d. 10

15. 13 Problem
The value of sin 3 5 7 is :
sin sin sin
16 16 16 16
2
a.
16
b. 1
8
1
c. 16
2
d. 32

16. 14 Problem
tan 3 1
If 3 , then the general value of is :
tan 3 1
n
a.
3 12
b. n 7
12
c. n 7
3 36
d. n
12

17. 15 Problem
sin B
In any triangle ABC, If cos A
2 sin C
, then :
a. a = b = c
b. c = a
c. a = b
d. b = c

18. 16 Problem
In a triangle ABC, if b + c = 2a and A = 600 then ABC is :
a. Equilateral
b. Right angled
c. Isosceles
d. Scalene

19. 17 Problem
The co-ordinates of the point which divides the join of the points (2, -1, 3) and
(4, 3, 1) in the ratio 3 : 4 internally are given by : .
2 20 10
a. , ,
7 7 7
10 15 2
b. , ,
7 7 7
20 5 15
, ,
c. 7 7 7
15 20 3
, ,
7 7 7
d.

20. 18 Problem
The area of the triangle ABC, in which a = 1, b = 2, C = 600, is :
a. 4 sq unit
1
b. 2 sq unit
c. 3 sq unit
2
d. 3 sq unit

21. 19 Problem
In a triangle ABC, b = , c = 1 and = 300 , then the largest angle of the triangle
is :
a. 600
b. 1350
c. 900
d. 1200

22. 20 Problem
If A + B + C = , then sin 2A + sin 2B + sin2C is equal to :
a. 4 sin A sin B sic C
b. 4 cos A cos B cos C
c. 2 cos A cos B cos C
d. 2 sin A sin B sin C

23. 21 Problem
A flag is standing vertically on a tower of height b. On a point at a distance a from
the foot of the tower, the flag and the tower subtend equal angles. The height of
the flag is :
a2 b2
a.
a2 b2
a2 b2
b.
a2 b2
c. a2 b2
a2 b2
a2 b2
d. a2 b2

24. 22 Problem
If are the roots of the equation 6x2 – 5x + 1 = 0, then the value of tan 1 tan 1
is
a. 0
b. /4
c. 1
d. /2

25. 23 Problem
The three stainght lines ax + by = c, b x + cy =a and c x + ay =b are collinear, if:
a. b+ c=a
b. c + a=b
c. a+ b + c=0
d. a + b =c

26. 24 Problem
The length of perpendicular from the point (a cos , a sin ) upon the straight
line y = x tan + c, c > 0, is :
a. c
b. c sin2
c. c cos2
d. c sec2

27. 25 Problem
The equation of the circumcircle of the triangle formed by the lines
x = 0, y = 0, 2x + 3y = 5 is :
a. 6(x2 + y2) + 5 (3x – 2y) = 0
b. x2 + y2 – 2x –3y + 5 = 0
c. x2 + y2 + 2x –3y - 5 = 0
d. 6(x2 + y2) - 5 (3x + 2y) = 0

28. 26 Problem
The differential equation of system of concentric circles with centre (1, 2) is :
a. (x - 2) + (y -1) dy = 0
dx
dy
b. (x - 1) + (y -2) dx
=0
dy
c. (x + 1) dx + (y - 2) = 0
dy
d. (x + 2) dx
+ (y -1) = 0

29. 27 Problem
The equation of pair of lines joining origin to the points of intersection of x2 +
y2 = 9 and x + y = 3 is :
a. x2 + (3 - x)2 = 9
b. xy = 0
c. (3 + y)2 + y2 = 9
d. (x - y)2 = 9

30. 28 Problem
The value of , for which the circle x2 + y2 + 2 x + 6y + 1 = 0 intersects the circle
x2 + y2 + 4x + 2y = 0 intersects the circle x2 + y2 + 4x + 2y = 0 orthogonally, is :
a. 11/8
b. -1
c. -5/4
d. 5/2

31. 29 Problem
The value of m, for which the line y = mx + 25 3 is a normal to the conic
x2 y2
3
1, is
16 9
2
a. 3
b. 3
c. 3
2
d. none of these

32. 30 Problem
The value of c, for which the line y = 2x + c is a tangent to the circle x2 + y2 =
16, is :
a. -16 5
b. 4 5
c. 16 5
d. 20

33. 31 Problem
The value of , for which the equation x2 – y2 – x + y – 2 = 0 represents a pair of
straight lines, are :
a. -3, 1
b. -1, 1
c. 3, -3
d. 3, 1

34. 32 Problem
The focus of the parabola x2 + 2y + 6x = 0 is :
a. (-3, 4)
b. (3, 4)
c. (3, -4)
d. (-3, -4)

35. 33 Problem
The value of m, for which the line y = mx + 2 becomes a tangent to the conic
4x2 – 9y2 = 36, are ;
a. 2
3
b. 2 2
3
8
c.
9
4 2
d. 3

36. 34 Problem
The eccentricity of the conic 4x2 + 16y2 – 24x – 32y = 1 is :
a. 1
2
b. 3
3
c. 2
3
d. 4

37. 35 Problem
The number of maximum normals which can be drawn from a point to ellipse
is :
a. 4
b. 2
c. 1
d. 3

38. 36 Problem
The equation of line of intersection of planes 4x + 4y – 5z = 12, 8x + 12y – 31z =
32 can be written as :
a. x 1 y 2 z
2 3 4
b. x 1 y 2 z
2 3 4
x y 1 z 2
c. 2 3 4
x y z 2
d.
2 3 4

39. 37 Problem
If a line makes angle , , , with four diagonals of a cube, then the value of
sin2 sin2 sin2 sin2 is :
a. 4/3
b. 8/3
c. 7/3
d. 1

40. 38 Problem
The equation of the plane, which makes with co-ordinate axes, a triangle with
its centroid , , , is :
a. x y z 3
b. x y z 1
x y z
c. =3
d. x y z =1

41. 39 Problem
If the points (1, 1), (-1, -1), (- 3, 3 ) are the vertices of a triangle, then
this triangle is :
a. Right-angled
b. Isosceles
c. Equilateral
d. None of these

42. 40 Problem
A variable plane moves so that sum of the reciprocals of its intercepts on the
co-ordinate axes is ½. Then the plane passes through :
1 1 1
, ,
a. 2 2 2
b. (-1, 1, 1)
c. (2, 2, 2)
d. (0, 0, 0)

43. 41 Problem
The direction cosines l, m, n of two lines are connected by the relations l +
m + n = 0, lm = 0, then the angle between them is ;
a. /3
b. /4
c. /2
d. 0

44. 42 Problem
     
The value of [a b ca b c] is :
  
a. [a b c ]
b. 0
  
c. 2 [a b c ]
  
d. a x (b x c )

45. 43 Problem
The area of the triangle having vertices as ˆ
i 2ˆ
j ˆ i
3k , ˆ 3ˆ
j ˆ i
3k , 4ˆ 7ˆ
j ˆ
7k , is
:
a. 36 sq unit
b. 0 sq unit
c. 39 sq unit
d. 11 sq unit

46. 44 Problem
The figure formed by the four points ˆ
i ˆ
j ˆ i
k , 2ˆ 3ˆ,5ˆ
j j ˆ ˆ
2k and k ˆ
j is :
a. Trapezium
b. Rectangle
c. Parallelogram
d. None of the above

47. 45 Problem
  
The equation of the plane passing through three non-collinear points a, b, c is :
      
a. r (b x c cxa a x b) 0
         
b. r (b x c cxa a x b) [a b c]
     
r (a x (b x c )) [a b c ]
c.
   
d. r (a b c) 0

48. 46 Problem
The unit vector perpendicular to and coplanar with is :
2ˆ
i 5ˆ
j
a. 29
b. 2ˆ 5 ˆ
i j
1
c. 2 (ˆ
i ˆ
j )
d. ˆ
i ˆ
j

49. 47 Problem
  2   2
(a x b) (a b) is equal to :
 
a. a2 b2
 
b. a2 b2
c. 1
 
d. 2 a2 b2

50. 48 Problem
The domain of the function f(x) = exp ( 5x 3 2x2 ) is :
a. [3/2, )
b. [1, 3/2]
c. (- , 1]
d. (1, 3/2)

51. 49 Problem
sin x
lim is equal to :
x x
a.
b. 1
c. 0
d. does not exist

52. 50 Problem
For the function which of the following is correct :
a. lim f ( x ) does not exist
x 0
b. lim f ( x) =1
x 0
c. lim f (x) exists but f(x) is not continuous at x = 0
x 0
d. f(x) is continuous at x = 0

53. 51 Problem
x (fo fo...of )(x)
If f ( x) , then is equal to :
x 1 19 times
x
a. x 1
19
b. x
x 1
c. 19 x
x 1
d. x

54. 52 Problem
A function f is defined by f(x) = 2 + (x - 1)2/3 in [0, 2]. What of the following is not
correct ?
a. f is not derivable in (0, 2)
b. f is continuous in [0, 2]
c. f(0) = f(2)
d. Rolle’s theoren is ture in [0, 2]

55. 53 Problem
2x 1
If f(x) = x 5
(x 5) , then f-1 (x) is equal to :
x 5 1
a. ,x
2x 1 2
b. 5x 1
,x 2
2 x
c. 5x 1
,x 2
2 x
x 5 1
d. ,x
2x 1 2

56. 54 Problem
d 1 1 x2 1 is equal to :
tan
dx x
1
a.
1 x2
x2
b.
2 1 x 2 ( 1 x 2 1)
c. 2
1 x2
1
d. 2 1 x2

57. 55 Problem
x is equal to :
1 cos
d 1 2
tan
dx x
1 cos
2
a. -1/4
b. 1/4
c. -1/2
d. 1/2

58. 56 Problem
The maximum value of x1/x is :
a. 1/ee
b. e
c. e1/e
d. 1/e

59. 57 Problem
The function f defined by f(x) = 4x4 – 2x + 1 is increasing for :
a. x < 1
b. x > 0
c. x < 1/2
d. x > 1/2

60. 58 Problem
A particle moves in a straight line so that s = t , then its acceleration is
proportional to :
a. (veloctiy)3
b. velocity
c. (velocity)2
d. (velocity)3/2

61. 59 Problem
32 x 2 (log x )2 dx is equal to :
a. 8x4(log x)2 + c
b. x4{8(log x)2 – 4(log x} + c
c. x4{8(log x)2 – 4 log x} + c
d. x3 {(log x)2 – 2 log x} + c

62. 60 Problem
cos x 1 x is equal to :
e dx
sin x 1
e x cos x
a. c
1 sin x
e x sin x
c
b. 1 sin x
ex
c
c. 1 sin x
e x cos x
c
d. 1 sin x

63. 61 Problem
If f(x) dx = g(x) + c, then f-1 (x) dx is equal to :
a. x f-1 (x) + c
b. f(g-1 (x)) + c
c. xf-1(x) – g (f-1 (x)) + c
d. g-1(x) + c

64. 62 Problem
The value of 2 dx is :
1 x(1 x 4 )
1 17
a. log
4 32
1 32
b. log
4 17
17
c. log
2
1 17
log
d. 4 2

65. 63 Problem
The value of the integral b x dx is :
a
x a b x
a.
1
b. 2
(b - a)
c. /2
d. b – a

66. 64 Problem
The area bounded by y = log x, x-axis and ordinates x = 1, x = 2 is :
a. (log 2)2
b. log 2/e
c. log 4/e
d. log 4

67. 65 Problem
The area of the segment of a circle of radius a subtending an angle of 2 at the
centre is :
1
a2 sin2
a. 2
1 2
b. a sin 2
2
1
c. a2
2
sin2
d. a2

68. 66 Problem
dy 2yx 1
The solution of the differential equation is :
dx 1 x2 (1 x 2 )2
a. y(1 + x2) = x + tan-1 x
y
b. 1 x2 = c + tan-1 x
c. y log (1 + x2) = c + tan-1 x
d. y (1 + x2) = c + sin-1 x

69. 67 Problem
The solution of the differential equation xdy – y dx = x2 y2 dx is :
2
a. x + x y 2 = cx2
b. y - x2 y 2 = cx
c. x - x2 y 2 = cx
d. y + x2 y 2 = cx2

70. 68 Problem
dy
The solution of the differential equation ex y
x 2e y
is :
dx
a. y = ex-y – x2e-y + c
1
b. ey - ex = 3 x3 + c
1
c. ex + ey = x3 + c
3
d. ex – ey =1 x3 + c
3

71. 69 Problem
If A and B are 2 x 2 matrices, then which of the following is true ?
a. (A + B)2 = A2 + B2 + 2AB
b. (A - B)2 = A2 + B2 – 2AB
c. (A - B) (A + B) = A2 + AB – BA – B2
d. (A + B) (A - B) = A2 – B2

72. 70 Problem
If M and N are any two events. The probability. That exactly one of them
occurs, is :
a. P(M) + P(N) – P(M N)
b. P(M) + P(N) + P(M N)
c. P(M) + P(N)
d. P(M) + P(N) – 2P(M N)

73. 71 Problem
If four dice are thrown together. Probability that the sum of the number
appearing on them is 13, is :
a. 35
324
5
b. 216
11
c. 216
11
d. 432

74. 72 Problem
If is the angle between two regression lines with correlation coefficient
, then :
2
a. sin 1
2
b. sin 1
2
c. sin 1
2
sin 1
d.

75. 73 Problem
The value of .0 , where 0.0 stands for the number 0.0373737 …., is :
a. 37/1000
b. 37/990
c. 1/37
d. 1/27

76. 74 Problem
2
a b a
If is an imaginary root of unity, then the value of 2
is :
b c b
2
c a c
a. a3 + b3 + c3
b. a2b – b2c
c. 0
d. a3 + b3 + c3 – 3abc

77. 75 Problem
If A = {x, y}, then the power set of A is ;
{xy, yx}
{ , x, y}
{ , {x}, {2y}}
{ , {x}, {y}, {x, y}}

78. FOR SOLUTIONS VISIT WWW.VASISTA.NET